Mathematics Asked on January 5, 2022

Let $k_1, k_2, k_3$ be natural non-negative numbers such that $k_1+k_2+k_3=K$. Let $n_1, n_2, n_3 in {0, ldots, N}$ and such that $n_1+n_2+n_3=N$.

**Calculate**

$$

S=sum_{(k_1, k_2, k_3): k_1+k_2+k_3=K, ,, n_1+n_2+n_3=N}k_1^{n_1}times k_2^{n_2} times k_3^{n_3}

$$

**My attempt:**

I am thinking on representing this sum as a chain of sums over each summand $k_j$. For example, the interior sum would be:

$

sum_{k_3=0}^{K-k_1-k_2}k_3^{n_3}.

$

Using Sums of p-th powers formula we can get $$sum_{k_3=0}^{K-k_1-k_2}k_3^{n_3}=frac{B_{n_3+1}(K-k_1-k_2+1)-B_{n_3+1}}{n_3}.$$ So, the sum $S$ would be represented as a product of these ratios with Bernoulli numbers $B_n$.

Is there a better way on computing/estimating from above sum $S$?

Get help from others!

Recent Questions

- How can I transform graph image into a tikzpicture LaTeX code?
- How Do I Get The Ifruit App Off Of Gta 5 / Grand Theft Auto 5
- Iv’e designed a space elevator using a series of lasers. do you know anybody i could submit the designs too that could manufacture the concept and put it to use
- Need help finding a book. Female OP protagonist, magic
- Why is the WWF pending games (“Your turn”) area replaced w/ a column of “Bonus & Reward”gift boxes?

Recent Answers

- Joshua Engel on Why fry rice before boiling?
- Lex on Does Google Analytics track 404 page responses as valid page views?
- Jon Church on Why fry rice before boiling?
- Peter Machado on Why fry rice before boiling?
- haakon.io on Why fry rice before boiling?

© 2024 TransWikia.com. All rights reserved. Sites we Love: PCI Database, UKBizDB, Menu Kuliner, Sharing RPP